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On Closed Ideals in Smooth Classes

✍ Scribed by Vincent Thilliez

Publisher
John Wiley and Sons
Year
2001
Tongue
English
Weight
294 KB
Volume
227
Category
Article
ISSN
0025-584X

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✦ Synopsis

We study closedness properties of ideals generated by real - analytic functions in some subrings (\mathcal{C}) of (C^{\infty}(\Omega)), where (\Omega) is an open subset of (\mathbb{R}^{n}). In contrast with the case (\mathcal{C}=C^{\infty}(\Omega)), which has been clarified by famous works of HΓΆrmander, Lojasiewicz and Malgrange, it turns out that such ideals are generally not closed when (\mathcal{C}) is an ultradifferentiable class. If (\mathcal{C}) is sufficiently regular and non-quasianalytic, and under the assumption that the real zero locus of the ideal reduces to a single point, we obtain a sharp sufficient condition of closedness, expressed in terms of the geometry of common complex zeros for the germs of the generators at this point. This condition is shown to be also necessary in dimension 2 , when the ideal is principal. Some related questions about rings of ultradifferentiable germs and about ultradistributions are discussed.

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